Bayesian Identification of Double-Seasonal Vector Autoregressive Models

Abstract

Identifying the autoregressive (AR) orders of a multivariate time series is a foundational step whose accuracy governs every downstream modelling and forecasting task. When the series exhibits two simultaneously operating seasonal periodicities, as is routinely observed in hourly electricity loads and intraday financial prices, no principled Bayesian identification framework currently exists. This paper addresses this gap by introducing a complete Bayesian order identification procedure for double-seasonal vector autoregressive (DSVAR) models. These DSVAR models are multivariate processes governed by a multiplicative triple autoregressive operator that jointly captures regular, first-seasonal, and second-seasonal dynamics. We treat the three order indices as unknown discrete parameters and derive closed-form expressions for the joint posterior probability mass function of the order triple. Two complementary prior specifications are considered: a conjugate matrix normal-Wishart prior and Jeffreys’ non-informative prior. The analysis is carried out under the assumption of symmetric, normally distributed errors, which ensures analytical tractability and allows for the posterior probabilities to be evaluated exactly for every admissible combination of orders. Specifically, under each prior, the posterior mass function reduces to explicit determinantal expressions that can be evaluated by a straightforward three-dimensional grid search. Monte Carlo experiments on various DSVAR processes confirm that the proposed technique achieves high identification accuracy even at moderate sample sizes across a range of parameter configurations and prior choices. The proposed Bayesian procedure is benchmarked against the standard Bayesian information criterion (BIC), consistently achieving higher correct identification rates across all cases. Empirical applications to hourly electricity load data from the Czech Republic and Germany, as well as to hourly solar radiation in Najran, Saudi Arabia, demonstrate the practical applicability of the Bayesian identification method.